Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T12:59:23.091Z Has data issue: false hasContentIssue false

MODEL COMPLETENESS OF O-MINIMAL FIELDS WITH CONVEX VALUATIONS

Published online by Cambridge University Press:  13 March 2015

CLIFTON F. EALY
Affiliation:
DEPARTMENT OF MATHEMATICS, WESTERN ILLINOIS UNIVERSITY, MACOMB, IL 61455, USAE-mail:cf-ealy@wiu.eduE-mail:j-marikova@wiu.eduURL: http://www.wiu.edu/users/cfe100/URL: http://www.wiu.edu/users/jm112
JANA MAŘÍKOVÁ
Affiliation:
DEPARTMENT OF MATHEMATICS, WESTERN ILLINOIS UNIVERSITY, MACOMB, IL 61455, USAE-mail:cf-ealy@wiu.eduE-mail:j-marikova@wiu.eduURL: http://www.wiu.edu/users/cfe100/URL: http://www.wiu.edu/users/jm112

Abstract

We let R be an o-minimal expansion of a field, V a convex subring, and (R0,V0) an elementary substructure of (R,V). Our main result is that (R,V) considered as a structure in a language containing constants for all elements of R0 is model complete relative to quantifier elimination in R, provided that kR (the residue field with structure induced from R) is o-minimal. Along the way we show that o-minimality of kR implies that the sets definable in kR are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adler, H., A geometric introduction to forking and thorn forking. Journal of Mathematical Logic, vol. 9 (2009), no. 1, pp. 120.CrossRefGoogle Scholar
Baisalov, Y. and Poizat, B., Paires de structures o-minimales, this Journal, vol. 63 (1998), no. 2, pp. 570–578.Google Scholar
Bröcker, L., Families of semialgebraic sets and limits, Real algebraic geometry (Coste, M., Mahe, L., and Roy, M. F., editors), Lecture Notes in Mathematics, vol. 1524, Springer, Berlin, Heidelberg, 1992, pp. 145162.Google Scholar
Bröcker, L., On the reductions of semialgebraic sets by real valuations, Recent advances in real algebraic geometry and quadratic forms (Jacob, W. B., Lam, T.-Y., and Robson, R. O., editors), Contemporary Mathematics, vol. 155, 1994, pp. 7595.Google Scholar
Cherlin, G. and Dickmann, M. A., Real closed rings. II. Mmodel Theory, Annals of Pure and Applied Logic, vol. 25 (1983), no. 3, pp. 213231.CrossRefGoogle Scholar
Chernikov, A. and Kaplan, I., Forking and dividing in ntp2 theories, this Journal, vol. 77 (2012), no. 1, pp. 1–20.Google Scholar
van den Dries, L, T-convexity and tame extensions II, this Journal, vol. 62 (1997), no. 1, pp. 14–34.Google Scholar
van den Dries, -L, Limit sets in o-minimal structures, o-minimal structures, Proceedings of the RAAG Summer School, Lisbon, 2003.Google Scholar
van den Dries, L and Lewenberg, A. H., t-convexity and tame extensions, this Journal, vol. 60 (1995), pp. 74–102.Google Scholar
van den Dries, L and Maříková, J., Triangulation in o-minimal fields with standard part map. Fundamenta Mathematicae, vol. 209 (2010), no. 2, pp. 133155.Google Scholar
Hasson, A. and Onshuus, A., Embedded o-minimal structures. Bulletin of the London Mathematical Society, vol. 42 (2010), no. 1, pp. 6474.Google Scholar
Hrushovski, E. and Pillay, A., On nip and invariant measures. Journal of the European Mathematical Society, vol. 13 (2011), no. 4, pp. 10051061.Google Scholar
Kaplan, I. and Usvyatsov, A., Strict independence in dependent theories, preprint.Google Scholar
Maříková, J., O-minimal fields with standard part map. Fundamenta Mathematicae, vol. 209 (2010), no. 2, pp. 115132.CrossRefGoogle Scholar
Maříková, J., O-minimal residue fields of o-minimal fields. Annals of Pure and Applied Logic, vol. 162 (2011), no. 6, pp. 457464.Google Scholar
Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. Journal of the American Mathematical Society, vol. 9 (1996), no. 4, pp. 10511094.CrossRefGoogle Scholar